Question 1

In the Black-Derman-Toy (BDT) model, short rates are distributed as

Accepted Answer

A)  Normal 
B)  Lognormal 
C)  Exponential 
D)  None of the above 
A)  Normal 
B)  Lognormal 
C)  Exponential 
D)  None of the above B

Question 2

A one-factor bond pricing model implies that interest-rates of all maturities are driven by a single source of stochastic randomness. For example the system of interest rates may be described by the following equation: $d r ( T ) = \alpha ( r ( T ) , T ) d t + \sigma ( r ( T ) , T ) d W , \quad \forall T$ where $T$ denotes the maturity of different rates. A single-factor model implies that

Accepted Answer

A)  All rates either move up together or all move down together. 
B)  The yield curve experience parallel shifts. 
C)  Instantaneous changes in rates of all maturities are perfectly positively or negatively correlated with each other. 
D)  Twists in shape of the yield curve are not possible. 
A)  All rates either move up together or all move down together. 
B)  The yield curve experience parallel shifts. 
C)  Instantaneous changes in rates of all maturities are perfectly positively or negatively correlated with each other. 
D)  Twists in shape of the yield curve are not possible. C

Question 3

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity cap on the one-year interest rate at a strike rate of 8% and a notional of $100?

Accepted Answer

A)  1.000 
B)  1.025 
C)  1.050 
D)  1.075 
A)  1.000 
B)  1.025 
C)  1.050 
D)  1.075 B

Question 4

Based on your answers to the previous two questions and a comparison of the prices of the cap and floor, what can you say about the forward rate between one and two years?

Accepted Answer

A)  The forward rate is roughly equal to the strike rate of the cap and floor. 
B)  The forward rate is such that the mark-to-market value of the FRA at the strike rate of the cap and floor will be zero. 
C)  Both (a) and (b). 
D)  There is not enough information to say anything about the forward rate. 
A)  The forward rate is roughly equal to the strike rate of the cap and floor. 
B)  The forward rate is such that the mark-to-market value of the FRA at the strike rate of the cap and floor will be zero. 
C)  Both (a) and (b). 
D)  There is not enough information to say anything about the forward rate.

Question 5

In the Cox-Ingersoll-Ross (1985) model, interest rates are specified by the following stochastic process: $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma _ {  } \sqrt { r _ { t } } d W _ { t }$ Implementation of the model to match observed nominal rate processes generally requires of the parameters that

Accepted Answer

A)  $k > 0$ 
B)  $\theta > 0$ 
C)  Both (a) and (b). 
D)  Neither (a) nor (b). 
A)  $k > 0$ 
B)  $\theta > 0$ 
C)  Both (a) and (b). 
D)  Neither (a) nor (b).

Question 6

In the Ho & Lee (1986) model, the parameter  $\delta$ plays a crucial role. Which of the following statements best describes this parameter?

Accepted Answer

A) $\delta > 1$ . B) As $\delta$ increases the volatility of interest rates increases. C) As $\delta$ increases the volatility of interest rates decreases. D) $\delta < 0$ . A) $\delta > 1$ . B) As $\delta$ increases the volatility of interest rates increases. C) As $\delta$ increases the volatility of interest rates decreases. D) $\delta < 0$ .

Question 7

In the Cox-Ingersoll-Ross (1985) model, interest rates are specified by the following stochastic process:  $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma _ {  } \sqrt { r _ { t } } d W _ { t }$ The process for interest rates is mean-reverting if

Accepted Answer

A) $k > 0$ B) $k 0$ D) $\theta < r _ { t }$ A) $k > 0$ B) $k 0$ D) $\theta < r _ { t }$

Question 8

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . At what strike price will one-year maturity call and put options on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon) have equal prices?

Accepted Answer

A)  $98.32 
B)  $99.52 
C)  $100.12 
D)  $101.42 
A)  $98.32 
B)  $99.52 
C)  $100.12 
D)  $101.42

Question 9

In the Black-Derman-Toy (BDT) model, short rates have

Accepted Answer

A)  Constant volatility for all maturities. 
B)  Volatility that changes by maturity of the short rate. 
C)  Volatility that varies by maturity and level of the short rate, i.e., state-dependent volatility. 
D)  Stochastic volatility. 
A)  Constant volatility for all maturities. 
B)  Volatility that changes by maturity of the short rate. 
C)  Volatility that varies by maturity and level of the short rate, i.e., state-dependent volatility. 
D)  Stochastic volatility.

Question 10

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity call option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

Accepted Answer

A)  0.80 
B)  0.90 
C)  1.00 
D)  1.10 
A)  0.80 
B)  0.90 
C)  1.00 
D)  1.10

Question 11

An exponential-affine short rate bond model is one

Accepted Answer

A)  That most bond traders have an affinity for. 
B)  Where the bond prices are linear in the short-rate. 
C)  Where the logarithm of bond prices is linear in the short rate. 
D)  Where the bond price is based on discrete compounding. 
A)  That most bond traders have an affinity for. 
B)  Where the bond prices are linear in the short-rate. 
C)  Where the logarithm of bond prices is linear in the short rate. 
D)  Where the bond price is based on discrete compounding.

Question 12

An affine factor model is one in which multiple factors  $X$ may be present. Which of the following is not true of an affine factor model.

Accepted Answer

A)  The drift $\mu ( X )$ will be linear in $X$ . 
B)  The volatility $\sigma ( X )$ will be linear in $X$ . 
C)  The yield $R ( X )$ will be linear in $X$ . 
D)  The logarithm of the price scaled by maturity is the yield. 
A)  The drift $\mu ( X )$ will be linear in $X$ . 
B)  The volatility $\sigma ( X )$ will be linear in $X$ . 
C)  The yield $R ( X )$ will be linear in $X$ . 
D)  The logarithm of the price scaled by maturity is the yield.

Question 13

In the Cox-Ingersoll-Ross or CIR model, interest rates are specified by the following stochastic process: $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ One attractive feature of this process relative to the Vasicek interest rate process $d r _ { t } = k \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ is that

Accepted Answer

A)  Interest rates are always non-negative in CIR while they may be negative in the Vasicek model. 
B)  There are parameter restrictions which guarantee non-negative stochastic interest rates in the CIR model, but there are no such restrictions possible in the Vasicek model. 
C)  It has extra parameters, so can fit observed yield curves better. 
D)  It allows for imperfect instantaneous correlation between rates of different maturities, whereas in the Vasicek model, they are perfectly correlated. 
A)  Interest rates are always non-negative in CIR while they may be negative in the Vasicek model. 
B)  There are parameter restrictions which guarantee non-negative stochastic interest rates in the CIR model, but there are no such restrictions possible in the Vasicek model. 
C)  It has extra parameters, so can fit observed yield curves better. 
D)  It allows for imperfect instantaneous correlation between rates of different maturities, whereas in the Vasicek model, they are perfectly correlated.

Question 14

In the Ho & Lee (1986) model, assume that the initial curve of zero-coupon discount bond prices for one and two years is  $0.9434$ and $0.8734$ , respectively. Assume that the probability of an upshift in discount functions is equal to that of a downshift. If the parameter $\delta = 0.95$ , then the price of a one-year zero-coupon bond in the up node after one year will be

Accepted Answer

A)  0.9282 
B)  0.9496 
C)  0.9563 
D)  0.9678 
A)  0.9282 
B)  0.9496 
C)  0.9563 
D)  0.9678

Question 15

Vasicek (1977) posits a general mean-reverting form for the short-rate:  $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ He then derives, in the absence of arbitrage, a restriction on the market price of risk $\lambda$ of any bond, where $( \mu - r ) / \eta = \lambda$ of any bond, with $\mu$ being the instantaneous return on the bond, and $\eta$ being the bond's instantaneous volatility. The derived restriction is that

Accepted Answer

A)  $\lambda$ is a constant. 
B)  $\lambda$ may be a function of time $t$ , but not of any other time- $t$ information or of the maturity $T$ of the bond. 
C)  $\lambda$ may be a function of the time- $t$ short rate $r _ { t }$ , but not of current time $t$ or of the bond maturity $T$ . 
D)  $\lambda$ may be a function of time $t$ and the time- $t$ short rate $r _ { t }$ , but not of the bond maturity $T$ . 
A)  $\lambda$ is a constant. 
B)  $\lambda$ may be a function of time $t$ , but not of any other time- $t$ information or of the maturity $T$ of the bond. 
C)  $\lambda$ may be a function of the time- $t$ short rate $r _ { t }$ , but not of current time $t$ or of the bond maturity $T$ . 
D)  $\lambda$ may be a function of time $t$ and the time- $t$ short rate $r _ { t }$ , but not of the bond maturity $T$ .

Question 16

In the Cox-Ingersoll-Ross (CIR 1985) model, you are given that $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ where $x = 0.5$ , $\theta = 0.06$ , $\sigma = 0.10$ . If the yield of a five-year bond is $0.07$ , then what is the price of the bond?

Accepted Answer

A)  0.65 
B)  0.70 
C)  0.75 
D)  0.80 
A)  0.65 
B)  0.70 
C)  0.75 
D)  0.80

Question 17

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity put option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

Accepted Answer

A)  1.00 
B)  1.08 
C)  1.16 
D)  1.24 
A)  1.00 
B)  1.08 
C)  1.16 
D)  1.24

Question 18

In the CIR (1985) model, which of the following statements is true? The price of the bond increases when

Accepted Answer

A)  The short rate $r _ { t }$ increases. 
B)  The rate of mean reversion $ { K }$ rises. 
C)  The long-run mean rate $\theta$ increases. 
D)  The volatility $\sigma$ increases. 
A)  The short rate $r _ { t }$ increases. 
B)  The rate of mean reversion $ { K }$ rises. 
C)  The long-run mean rate $\theta$ increases. 
D)  The volatility $\sigma$ increases.

Question 19

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity floor on the one-year interest rate at a strike rate of 8% and a notional of $100?

Accepted Answer

A)  1.000 
B)  1.026 
C)  1.052 
D)  1.078 
A)  1.000 
B)  1.026 
C)  1.052 
D)  1.078

Question 20

In the Vasieck (1977) model, you are given that $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma d W _ { t }$ where $k = 0.5$ , $\theta = 0.06$ , $\sigma = 0.10$ , and the current short rate of interest is $r _ { 0 } = 0.08$ . What is the expected standard deviation of the short rate of interest one year hence?

Accepted Answer

A)  0.08 
B)  0.09 
C)  0.10 
D)  0.11 
A)  0.08 
B)  0.09 
C)  0.10 
D)  0.11

Exam 27: Factor Models of the Term Structure

In the Black-Derman-Toy (BDT) model, short rates are distributed as

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be σ=0.30\sigma = 0.30σ=0.30 . What is the price of a one-year maturity cap on the one-year interest rate at a strike rate of 8% and a notional of $100?

Based on your answers to the previous two questions and a comparison of the prices of the cap and floor, what can you say about the forward rate between one and two years?

In the Ho & Lee (1986) model, the parameter δ\deltaδ plays a crucial role. Which of the following statements best describes this parameter?

In the Black-Derman-Toy (BDT) model, short rates have

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be σ=0.30\sigma = 0.30σ=0.30 . What is the price of a one-year maturity call option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

An exponential-affine short rate bond model is one

An affine factor model is one in which multiple factors XXX may be present. Which of the following is not true of an affine factor model.

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be σ=0.30\sigma = 0.30σ=0.30 . What is the price of a one-year maturity put option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

In the CIR (1985) model, which of the following statements is true? The price of the bond increases when

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be σ=0.30\sigma = 0.30σ=0.30 . What is the price of a one-year maturity floor on the one-year interest rate at a strike rate of 8% and a notional of $100?

Overview

Futures Markets

Pricing Forwards and Futures I

Pricing Forwards Futures II

Hedging With Futures Forwards

Interest-Rate Forwards Futures

Options Markets

Options: Payoffs Trading Strategies

No-Arbitrage Restrictions

Early-Exercise Put-Call Parity

Option Pricing: an Introduction

Binomial Option Pricing

Implementing the Binomial Model

The Black-Scholes Model

Mathematics of Black-Scholes

Beyond Black-Scholes

The Option Greeks

Path-Independent Exotic Options

Exotic Options II: Path-Dependent Options

Value at Risk

Swaps and Floating Rate Products

Equity Swaps

Currency and Commodity Swaps

Term Structure of Interest Rates: Concepts

Estimating the Yield Curve

Modeling Term Structure Movements

The Heath-Jarrow-Morton HJM and Libor Market Model LMM

Credit Derivative Products

Structural Models of Default Risk

Reduced-Form Models of Default Risk

Modeling Correlated Default

Filters

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity cap on the one-year interest rate at a strike rate of 8% and a notional of $100?

In the Ho & Lee (1986) model, the parameter $\delta$ plays a crucial role. Which of the following statements best describes this parameter?

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity call option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

An affine factor model is one in which multiple factors $X$ may be present. Which of the following is not true of an affine factor model.

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity put option on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon)?

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What is the price of a one-year maturity floor on the one-year interest rate at a strike rate of 8% and a notional of $100?